Seminars (MOSW02)

The first part of the talk will be devoted to work in progress together with O. Biquard and Ó. García-Prada on a version of parabolic Higgs bundles which correspond to (semistable) G-local systems on a punctured Riemann surface, where G is a real semisimple Lie group. In the second part we will talk about ongoing work with M. Logares on a generalization of parabolic bundles which correspond (via a construction which relates them to conic bundles) to Sp(6,R)-local systems on compact Riemann surfaces. Emphasis will be on the stability conditions; a unifying theme will be the interpretation of the local terms in the formula for the parabolic degree in terms of GIT.

We will discuss some generalities of the geometry, topology and singularities of the the G-character variety of F, that is, the moduli space Hom(F,G)/G of representations of a finitely presented group F into a Lie group G.

Then, we concentrate on the case when G is a complex affine reductive Lie group with maximal compact subgroup K, and F is a free group of rank r. In this situation, it can be proved that Hom(F,K)/K is a strong deformation retract of Hom(F,G)/G; in particular, both spaces have the same homotopy type. In the case G=SL(n,C), one can explicitly describe the singular locus of these character varieties, showing that they have the homotopy type of a manifold only when F or G are abelian, or r+n

This is a report on a joint work in progress with D.Arinkin. We investigate the self-duality question for the compactified stacky Jacobian of a reducible curve with planar singularities. This is needed for extending the classical limit Langlands duality for GL(n) to the whole moduli of Higgs bundles. I will explain how the self-duality statement extends to a single reduced but possibly reducible curve and will discuss the technical issues that one needs to overcome to prove the statement in families. I will also discuss the way self-duality interacts with stability.

I will discuss recent generalizations of the techniques of Atiyah and Bott on equivariant Morse theory. These extend results on stable bundles to Higgs bundles and stable pairs. As a consequence, information is obtained on the topology of representation varieties into noncompact Lie groups.

Let M be a closed surface of genus at least two, N a manifold of non-positive Hermitian curvature (the Siu-Sampson condition) and fix a homotopy class of maps from M to N (or a representation of the fundamental group of M in the group of isometries of N). For each complex structure J on M there is a harmonic map f:M->N (or an equivariant harmonic map of the universal covers). In situations where this map is unique it depends smoothly on J and its energy E defines a smooth function on the Teichmüller space of M. We prove that this function is plurisubharmonic, and study conditions when it is strictly plurisubharmonic.

This result was suggested by Gromov as an alternative way of developing and strengthening the Siu-Sampson rigidity theory. Indications of these applications will be given as time permits.

This talk will survey the methods and applications of our joint work with Katzarkov Pantev and Ramachandran arxiv/0409.0693.

In this talk we will present a structure theorem concerning causal representations; in particular we will discuss the rationality of the Toledo invariant in the non compact case and explain its relation to the characterisation of non tube type domains in terms of the hermitian triple product.

A convex real projective 3-orbifold is the quotient orbifold of a convex domain in $RP^3$ by a discrete group of projective automorphisms in $PGL(4, R)$. Hyperbolic 3-orbifolds form a subclass. The convex real projective 3-manifolds were begun to be studied by Cooper, Long, and Thistlethwaite. We will summarize some of the recent results on deforming convex real projective structures on 3-dimensional orbifolds, including those of Benoist, myself, Marquis, Lee, Hodgson, Cooper, Tillman, and so on. In particular, a numerical study of real projective structures on Coxeter orbifolds is included. Finally, we discuss open problems in this area. Our topic is related to understanding the deformations of $SL(4,R)$-representations of discrete groups.